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Why is the elastic limit different from the proportional limit for some materials?

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Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).
mechanics of solids Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).

The elastic limit and the proportional limit are distinct concepts in the mechanics of solids, though they may coincide for some materials.

The proportional limit marks the end of the range where a material obeys Hooke’s law, meaning the stress is directly proportional to the strain. This region is referred to as linearly elastic behavior.12

The elastic limit represents the maximum stress a material can withstand before undergoing permanent, or plastic, deformation. When stress remains within this limit, the material will return to its original shape once the force is removed.12

For some materials, these two limits are nearly identical and are often used interchangeably. However, for other materials, there exists a region of nonproportional elasticity between the proportional limit and the elastic limit. In this region, the material continues to behave elastically—meaning it will still recover its original shape—but the relationship between stress and strain is no longer linear.12